# I.B. Maths portfolio type 1 Matrices

IB MATHS SL                PANJI WICAKSONO

MATHS PORTOFOLIO SL TYPE 1: MATRIX POWER

CANDIDATES NAME: PANJI WICAKSONO

IB NOVEMBER 2008

INTRODUCTION:

Matrices are an array of numeric or algebraic quantities subject to mathematical operations that can be multiplied, subtracted or added. The numbers can be arranged in a rectangular array of numbers set out in rows and columns. The numbers that are inside the matrices are called entries. Matrices are useful to keep track of the coefficients of linear transformations and to record data that depend on multiple parameters. In a more complex use, it can be helpful in encrypting numerical data and also in computer graphics.

In this portfolio, the task that was given was to deduce a general formula or pattern of the given matrices and also to determine the limitation of the general formula that has been found. This task will be done using the help of a GDC calculator (GDC-TI 83 plus) and the knowledge that have been attained during the mathematic SL course.

QUESTION 1:

Consider the matrix

Calculate Mn for n = 2, 3, 4, 5, 10, 20, 50

Describe in words any pattern you observe.

Use this pattern to find a general expression for the matrix Mn in terms of n

Counting for n= 2 manually, it gives:

M2 = M.M = =

For n= 3
M3 = M2.M =  =

For n = 4

M4 = M3.M = =

For n = 5

M5= M4.M =  =

For the larger values of n, the GDC was used to help ease the task by first pressing MATRIX menu EDIT option, and then chose matrix 1: [A] and defined the correct order of the matrix (dimensions), 2 x 2. Then I entered the matrix entries of and save it to memory 1 slot.  Then the matrix was used for calculations by pressing MATRIX menu, NAMES submenu and raised it to the power I was calculating.

Using the GDC to calculate n = 10, 20, 50 the result was:

n = 10

M10 =  =  =

n = 20

M20 =  = =

n = 50

M50 =     =

By calculating the values that were given in question number 1, I noticed a pattern that occurs from one value of n to the other. The 2 inside the matrix are always raised to the value of n in Mn. The result gathered was that in entries ‘a’ and ‘d’ which are number greater than one, follows this pattern:

Mn =

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